1.Only certain DSi flashcarts work, and they have to be updated to work with the 3DS
2. No and yes, but you can't play them.
3.By DLC games I take it you really mean Downloadable titles, and there are, but unless you're an ambassador (meaning you got it before the middle of last month) you won;t be getting any GBA games (as far as is planned), and there are no downloadable DS games.

1. no, flash cards for the fat ds and ds lite won't work on dsi or 3ds. the flash carts for dsi will work on 3ds.
2. no flash cart for 3ds and IIRC 3 3ds games are dumped, but that doesn't mean anything.
3. no gba or ds games on eshop, gba probably eventually. as of now Nintendo has no plans to sell gba games on eshop

1. Acekard and the DSTWO are two working flashcarts. You need to have them updated to the latest firmware, though.
2. As of yet, there is no such thing as a flashcard for 3DS roms. The 3DS isn't even hacked.
3. Do you mean VC games? They only have NES and Gameboy games for download. (unless you're an ambassador, you won't be able to download any GBA games on eshop)

1. do the original ds flash carts work with it?
2. are there 3ds flash carts and roms?
3. is there dlc games like gba games and ds games?

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1. Cards like the DSTWO work with it when updated to a working firmware.
2. There are no 3ds flash cards atm. Supposedly, there are a few 3ds roms floating around, but no one knows if they are working roms because there isn't a way to test them.
3. Current DLC comes as GB/C and 3DSWare. Select NES games atm and soon-to-be GBA games are only accessible by Ambassadors that bought and registered their device before the August deadline.